Slope of the perpendicular line on the other line

In summary: Another approach is to use the fact that perpendicular lines have slopes that are negative reciprocals of each other (m*m' = -1). This can be shown using the slope formula (m = (y2 - y1)/(x2 - x1)), where the two points are a point on one line and a point on the other line. From there, you can use algebra to solve for m' to get m' = -1/m. In summary, the slope of a perpendicular line to another line is equal to the negative reciprocal of the slope of the original line. This can be proved using geometric principles or the slope formula and algebra.
  • #1
Alg0r1thm
11
0
Why is the slope of perpendicular line on the the other defined this way:
m = slope of a particular line
m'= slope of the perpendicular line on that particular line

m*m' = -1 OR m' = -1/m

Thanks
 
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  • #2
If you have a line y=mx+c, and another line y'=m'x+c', what does m' have to be for y' to be perpendicular to y.
Draw a few lines and see.

i.e. if m=m', then the two lines are parallel.
 
  • #3
Simon Bridge said:
If you have a line y=mx+c, and another line y'=m'x+c', what does m' have to be for y' to be perpendicular to y.
Draw a few lines and see.

i.e. if m=m', then the two lines are parallel.

You mean there is no kind of proof or something like algebraic statement which states that the perpendicular line slope would be concluded via it?
 
  • #4
One way to look at it is this: if y= mx+ b is the equation of a line, then the slope, m, is the tangent of the angle, [itex]\theta[/itex], the line makes with the x-axis. If two lines are perpendicular then they form a right triangle with the x-axis as hypotenuse. The angle one of the lines makes with the x- axis, say [itex]\theta[/itex], will be acute, the other, [itex]\phi[/itex], will be obtuse.

Looking over this I see I have used the wrong words. I meant that one will be less that or equal to 45 degrees, the other larger than or equal to 45 degrees.

The angles inside that right triangle will be [itex]\theta[/itex] and [itex]\pi- \phi[/itex] and we must have [itex]\theta+ (\pi- \phi)= \pi/2[/itex] so that [itex]\phi- \theta= \pi/2[/itex].

Now use [itex]tan(a+ b)= \frac{tan(a)+ tan(b)}{1- tan(a)tan(b)}[/itex] with [itex]a= \phi[/itex] and [itex]b= -\theta[/itex]:
[tex]tan(\phi- \theta)= \frac{tan(\phi)- tan(\theta)}{1+ tan(\phi)tan(\theta)}= tan\left(\frac{\pi}{2}\right)[/tex]

But [itex]tan(\pi/2)[/itex] is undefined! We must have the fraction on the left undefined which means the denominator must be 0: [itex]1+ tan(\theta)tan(\phi)= 0[/itex] so that [itex]tan(\theta)tan(\phi)= -1[/itex] and that last is just "[itex]mm'= -1[/itex]".
 
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  • #5
Alg0r1thm said:
You mean there is no kind of proof or something like algebraic statement which states that the perpendicular line slope would be concluded via it?
NO - I mean that you can start from what I wrote and work it out for yourself.
I was trying to set your feet on the right path.

HallsofIvy showed you one approach.
 
1.

What is the slope of a perpendicular line on another line?

The slope of a perpendicular line on another line is the negative reciprocal of the original line's slope. This means that if the original line has a slope of m, the perpendicular line will have a slope of -1/m.

2.

How do you find the slope of a perpendicular line?

To find the slope of a perpendicular line, you can use the formula -1/m, where m is the slope of the original line. Alternatively, you can also use the slope formula to find the slope of two points on the perpendicular line and compare it to the slope of the original line.

3.

Can a perpendicular line have the same slope as the original line?

No, a perpendicular line can never have the same slope as the original line. This is because the slope of a perpendicular line is always the negative reciprocal of the original line's slope, meaning it will always be a different value.

4.

What is the significance of the slope of a perpendicular line?

The slope of a perpendicular line is important in geometry and trigonometry, as it helps determine the relationship between two lines. It can also be used to find the angle between two intersecting lines.

5.

How can the slope of a perpendicular line be used in real-world applications?

The concept of perpendicular slopes is used in various fields such as architecture, engineering, and physics. For example, architects use perpendicular lines to create right angles and ensure the stability of structures. In physics, perpendicular slopes are used to calculate the force of an object sliding down a ramp.

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